No need to panic.
Products like quick start are marketed as an instant cycle, and as you have found out that isnt the case. At best they will speed up the cycling process, at worst they do nothing. Your friend was right to suggest using such a product as it can help, but to think it will cycle a tank overnight is a bit naive.
The fishkeeping industry is largely unregulated so company's that sell these products can pretty much say what they want. Also, you can add fish right away if you use these products, its not a lie. I dont think the claim is for an instant cycle, its just you can add fish. You can also add fish right away if you dont add these products. In both cases however you need to go through a fish in cycle process. If you use a bottled bacteria product it might speed up this fish in cycle from months to weeks though.
Is your fish fine? Yes. So no need to worry. The levels of waste you are seeing is relatively safe and a normal part of the cycle process.
Your target should be to keep ammonia + nitrite below 0.5ppm combined. If you see for example 0.25ppm of ammonia and 0.25ppm of nitrite do a small 25% water change. You need some ammonia and nitrite to fuel the cycle. Eventually (3 to 8 weeks typically) you wont need to do water changes to maintain 0 ammonia and nitrite and can cut back to doing them to control nitrate, typically to keep nitrate below 40ppm.
You are doing fine. No need to move the fish into a smaller tank, also uncycled. If anything you might be doing too many water changes, you are removing waste before your beneficial bacteria has had chance to use it to grow. Low levels of ammonia and nitrite is relatively safe as long as you don't allow them to elevate, and necessary to the cycle process.